Simplify the Matrix

Problem

[[1,2,1,0,0,20],[1,1,0,1,0,12],[5,1,0,0,1,40],[−20,−30,0,0,0,0]]

Solution

1. Identify the pivot in the second column to begin Gauss-Jordan elimination, choosing the element at (R_1),(C_2) Divide the first row by 2 to create a leading 1.

1/2*(R_1)→(R_1)

[[0.5,1,0.5,0,0,10],[1,1,0,1,0,12],[5,1,0,0,1,40],[−20,−30,0,0,0,0]]

2. Eliminate the other entries in the second column by performing row additions.

(R_2)−(R_1)→(R_2)

(R_3)−(R_1)→(R_3)

(R_4)+30*(R_1)→(R_4)

[[0.5,1,0.5,0,0,10],[0.5,0,−0.5,1,0,2],[4.5,0,−0.5,0,1,30],[−5,0,15,0,0,300]]

3. Identify the next pivot in the first column. We use the element at (R_2),(C_1) Multiply the second row by 2 to create a leading 1.

2*(R_2)→(R_2)

[[0.5,1,0.5,0,0,10],[1,0,−1,2,0,4],[4.5,0,−0.5,0,1,30],[−5,0,15,0,0,300]]

4. Eliminate the other entries in the first column.

(R_1)−0.5*(R_2)→(R_1)

(R_3)−4.5*(R_2)→(R_3)

(R_4)+5*(R_2)→(R_4)

[[0,1,1,−1,0,8],[1,0,−1,2,0,4],[0,0,4,−9,1,12],[0,0,10,10,0,320]]

5. Swap rows to organize the identity matrix structure (Reduced Row Echelon Form).

(R_1)⇔(R_2)

[[1,0,−1,2,0,4],[0,1,1,−1,0,8],[0,0,4,−9,1,12],[0,0,10,10,0,320]]

Final Answer

[[1,2,1,0,0,20],[1,1,0,1,0,12],[5,1,0,0,1,40],[−20,−30,0,0,0,0]]=[[1,0,−1,2,0,4],[0,1,1,−1,0,8],[0,0,4,−9,1,12],[0,0,10,10,0,320]]

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